In this section we provide the proofs of the results needed for establishing the asymptotic normality of the estimators of the entries of \(^*\) .
The results below make use of the fact that consistent clustering is possible, under our assumptions, as stated below.

Let \(=: \lbrace \hat{G}=G^*\rbrace \) , for \(\hat{G}\)  estimated by either the COD or the PECOK algorithm of [1]}.
Then, under Assumptions REF  and REF , we have
\(()\ge 1-\frac{C}{(d\vee n)^3}.\) 
